Problem

Calculate the range of commute distances covering approximately 68% of the students: At one university, the mean distance commuted to campus by students is 19.0 miles, with a standard deviation of 4.1 miles. Suppose that the commute distances are normally distributed. Complete the following statements. (a) Approximately $68 \%$ of the students have commute distances between $\square$ miles and $\square$ miles. (b) Approximately ? $\quad 0$ of the students have commute distances between 6.7 miles and 31.3 miles.

Solution

Step 1 :The problem is asking for the range of commute distances that cover approximately 68% of the students. In a normal distribution, approximately 68% of the data falls within one standard deviation of the mean. Therefore, to find the range, we need to add and subtract the standard deviation from the mean.

Step 2 :Given that the mean distance commuted to campus by students is 19.0 miles, and the standard deviation is 4.1 miles.

Step 3 :To find the lower bound of the range, subtract the standard deviation from the mean: \(19.0 - 4.1 = 14.9\) miles.

Step 4 :To find the upper bound of the range, add the standard deviation to the mean: \(19.0 + 4.1 = 23.1\) miles.

Step 5 :Final Answer: Approximately 68% of the students have commute distances between \(\boxed{14.9}\) miles and \(\boxed{23.1}\) miles.

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Source: https://solvelyapp.com/problems/7PyOwkBK9Q/

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